We will factorize the quadratic equation \( 20x^2 + 47x + 24 = 0 \) by the method of splitting the middle term.

The equation is of the form \( ax^2 + bx + c = 0 \), where:

- \( a = 20 \)
- \( b = 47 \)
- \( c = 24 \)

### Step 1: Multiply \( a \) and \( c \).
\[
a \cdot c = 20 \cdot 24 = 480
\]

### Step 2: Find two numbers that multiply to 480 and add to 47.
The two numbers are 32 and 15 because:

- \( 32 \cdot 15 = 480 \)
- \( 32 + 15 = 47 \)

### Step 3: Split the middle term using these numbers.
\[
20x^2 + 32x + 15x + 24 = 0
\]

### Step 4: Factor by grouping.
Group the terms:
\[
(20x^2 + 32x) + (15x + 24) = 0
\]

Factor each group:
\[
4x(5x + 8) + 3(5x + 8) = 0
\]

### Step 5: Factor out the common binomial factor.
\[
(4x + 3)(5x + 8) = 0
\]

### Step 6: Solve for \( x \).
Set each factor equal to zero:
\[
4x + 3 = 0 \quad \text{or} \quad 5x + 8 = 0
\]
Solving these:
\[
x = -\frac{3}{4} \quad \text{or} \quad x = -\frac{8}{5}
\]

Thus, the factorized form of the equation is:
\[
(4x + 3)(5x + 8) = 0
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve a quadratic equation using the quadratic formula?
2. Can all quadratic equations be factorized?
3. What is the method of completing the square?
4. How do you graph the quadratic equation \( 20x^2 + 47x + 24 = 0 \)?
5. How does the discriminant help in determining the nature of the roots?

### Tip:
When factorizing, if you can't easily find numbers to split the middle term, consider using the quadratic formula.

Learn how to factorize the quadratic equation 20x² + 47x + 24 using the method of splitting the middle term. This problem covers key algebraic concepts, including factoring and solving quadratic equations, and is suitable for students in Grades 9-10.

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